I would highly recommend doing a serpentine stitch rather than a straight stitch, using a lightweight variegated thread (such as Aurifil 50wt). For this you would be sewing lines from edge to edge, going over the seams yes, but also extending the seams into the borders. It takes more thread to do a serpentine stitch, but it is ***much*** easier to sew and the serpentine is a very forgiving stitch (unlike the straight stitch). A serpentine stitch is like a wavy straight stitch. Aside from being easy to use, I like the serpentine for the softness it gives to the quilting.
I would do the serpentine top/bottom and side-to-side, going over each edge of the sashing. Extend the seam lines into the border by marking with a ruler and sliver of white soap, if necessary. I would also sew serpentine lines that would cover all of the edges of the inner squares. In other words, you would end up with a grid of squares all over the quilt, but they would be sewn in straight lines to cover seams. Everything is sewn in long straight lines, going from one edge to the other edge.
There are lots of things you can do to make the process easier with a better end product. Heavily starching the backing fabric before layering is a good idea, as it greatly helps to prevent puckers. Splitting the batting into thirds reduces bulk under the arm; it would be a good idea to research this option before starting. Spray basting is better than other types of basting for this type of work, as it keeps the layers together better. Any time you are sewing a grid, you have the risk of sewing in puckers when you start crossing lines. Starch and spray basting help a lot, but you also have to be careful not to stretch the fabric out of shape as you sew. Creating a flat styrofoam surface around your sewing machine helps a lot with that (there are Youtube videos that show how to create this inexpensive table).
If you can, it would be a good idea to make a mock-up quilt out of cheap muslin first to practice on, so you are sure of what you are doing before you tackle the quilt.
As for SID, I had refused to do it long before I tried the serpentine stitch. I find SID very frustrating because I notice every little deviation from the seamline, plus it gives me eye strain and anxiety attacks while sewing! The serpentine stitch, on the other hand, I actually enjoy.